Equilibrium and efficient clustering of arrival times to a queue

0.25 MB·

by  Amihai Glazer

Equilibrium and efficient clustering of arrival times to a
queue
7 Amihai Glazer∗, Refael Hassin†and Liron Ravner‡
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January 18, 2017
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Abstract

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We consider a game of decentralized timing of jobs to a single server (machine)
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with a penalty for deviation from some due date and no delay costs. The jobs sizes
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s are homogeneous and deterministic. Each job belongs to a single decision maker, a
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customer,whoaimstoarriveatatimethatminimizeshisdeviationpenalty. Ifmultiple
[

customers arrive at the same time then their order is determined by a uniform random
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v draw. If the cost function has a weighted absolute deviation form then any Nash
6 equilibriumis pureandsymmetric, that is, all customers arrive together. Furthermore,
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weshowthatthereexistmultiple, infactacontinuum, ofequilibriumarrivaltimes, and
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4 providenecessaryandsufficientconditionsforthesocially optimalarrivaltimetobean
0 equilibrium. Thebasemodelissolvedexplicitly, buttheprevalenceofapuresymmetric
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1 equilibrium is shown to be robust to several relaxations of the assumptions: inclusion
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of small waiting costs, stochastic job sizes, random sized population, heterogeneous
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due dates and non-linear deviation penalties.
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:
v
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X 1 Introduction
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a
We study the equilibrium and optimal arrival time schedules to a single server system in
which customers incur a cost for deviation from their due date. The number of customers is
finite; each has a single job that requires processing. When some customers arrive together
they are ordered according to a uniform random draw. If all customers have a common
due date, or close due dates, then arriving together with other customers is potentially
beneficial because in expectation every customer is in the “middle,” which may be closer to
the due date than arriving before or after everyone else. If the job sizes of the customers
are deterministic and the cost function is a weighted absolute deviation, i.e. some linear
penalty for early service and a possibly different linear penalty for late service, we find that
∗Department of Economics, University of California, Irvine, [e-mail]
†Department of Statistics and Operations Research, Tel Aviv University, [e-mail]
‡Department of Statistics and Operations Research, Tel Aviv University, [e-mail]
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simultaneous arrivals (clustering) is the only possible equilibrium. The equilibrium arrival
time is not unique, but rather there is a whole interval such that if all customers arrive at
any time within that interval then no customer has an incentive to deviate. Moreover, the
socially optimal arrival time often lies within this interval and is therefore an equilibrium.
Specifically this holds if the earliness and tardiness penalties are not too different; we will
present simple explicit necessary and sufficient conditions.
This work can be seen as the game theoretic compliment to the classical machine tim-
ing/sequencing problem with a common due date and earliness and tardiness penalties. A
general framework for single server and weighted penalties is presented and analysed in [17].
Surveys of the research on these problems can be found in [7] and [19]. The literature has
focused on centralized analysis, that is, a single decision maker determining the schedule for
all jobs. We analyse this model in a decentralized setting in which each job is a rational
agent, where the centralized socially optimal schedule is a benchmark for comparison. This
work also relates to the research on decentralized multi-machine routing games ([10],[12] and
[6]), where the decision variable for the individual jobs is the choice of machine and not a
timing decision.
Our base model assumes linear penalties, deterministic population and job sizes, no wait-
ing costs and common due dates, and is thus very tractable and yields explicit results. The
equilibrium solution of simultaneous arrivals, however, is valid for a more general decen-
tralized job timing game. Specifically the due dates need not be common, but rather close
enough, the cost function needs to be unimodal, there may be small waiting costs and the
job sizes and population sizes may be random. We explore all of these possibilities and also
present examples for which a symmetric pure-strategy equilibrium does not exist.
The next section reviews relevant literature. Section 3 defines the base model. This is
followed byderiving thesociallyoptimal solutioninSection 4, equilibrium analysis inSection
5 and sensitivity analysis in Section 6.
2 Literature
Our paper considers the time at which a person chooses to join a queue, with a first-come
first-served order of service. In our model a customer does not care how long he stays in the
queue, but does care about the time at which he is served. Even if a customer does not care
about when he is served, he may care about when he joins the queue because that affects
his waiting time. An arrival-time game to a discrete stochastic queue first appeared in [15],
which introduced the ?/M/1 model: each customer chooses when to arrive at a single-server
queue that starts operating at some known time, aiming to minimize his minimizing waiting
costs. A related problem concerns the concert queuing game, where customers seek service
at a first-come first-served queueing system that opens up at a given time, with a customer’s
utility depending on when he gets service and how long he waits in the queue [23]. For
a server with no defined starting time or ending time, customers who care only about the
length of time they are in the queue will sort themselves out as much as possible [25]. This
model has been extended in many directions: batch service [16], loss systems [31, 21], no
early arrivals [18], and a network of queues [22]. We find that when customers instead care
about the time at which they get served, they will concentrate their arrival times.
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Queueing models which consider the timing of arrivals have incorporated tardiness penal-
ties, [23,20],orderpenalties[35],andearlinesspenalties[38]. In[36]adeterministicprocessor
sharingsystemwithheterogeneousduedatesisconsidered, andthepure-strategyequilibrium
arrival times are derived, explicitly for some examples, and algorithmically for the general
case. The latter model resembles ours in having customers who arrive together “helping”
each achieve his due date, however there is no clustering of arrivals. The arrival game to a
last-come-first-served system is studied by [34], who experimentally compare several service-
order policiesprovided in[8]. Akey featureinalloftheaboveisthattheequilibrium solution
is given by a mixed strategy, that is, customers randomize their arrival times. Furthermore,
the arrival distribution has no atoms (except for boundary cases) because arriving together
with other customers is never desirable.
Work in transportation (which rarely cites the work on queues) considers a bottleneck,
such asabridge, withacommuter’s costsincreasing withthetimehespends ontheroad, and
incurring a cost if he arrives at his destination too early or too late ([40] and [4]). The choice
of departure time in transportation is analyzed in [5], who introduce a model with masses
of customers departing together. They assume a fluid population with the Greenshield’s
congestion dynamics (e.g. [29]), which are, at least from a technical perspective, close to the
dynamics of processor sharing queue, in the sense that all customers travelling together are
slowed down as congestion increases.
Workineconomics has, like us, addressed issues of timing, but unlike usdoesnot consider
a server who serves customers in the order in which they arrive. A concern in this literature
is whether agents will cluster, all taking action at the same time; another concern has been
whether an agent benefits from acting before or instead after others do. Some authors show
how the first entrant into an industry can earn larger profits than later entrants, a result
called the pioneer’s advantage [37], the first-mover advantage [28], and order of market entry
effect [24]. Arriving last is desirable in the advertisement board timeline game of [1], where
the last ad posted is the most visible.
Researchers have also considered payoffs which depend on the order of arrivals. Delay
may be costly, with each player preferring that others act before him. Maynard Smith (1974)
[39] formalized such a situation, where two animals fight over a fallen prey, with the first
to give up losing, and with fighting costly for both. The situation may be reversed,with
the passage of time exogenously beneficial, and players wishing to pre-empt others. In the
“grab-the-dollar” game a player can either grab the money on the table or wait for one more
period, with the pot increasing over time. Each player wants to be the first to take the
money, but would rather grab a larger pot. The idea has been applied to firms’ decisions
about when to adopt a new technology [13], and about when to enter a market [14]. When a
firm should make an irreversible investment, where the return per period decreasing with the
number of firms that have invested, and with the cost of investment decreasing over time, is
modelled by [3]. A result is that the firms may cluster, even in in the absence of coordination
failures, informational spillovers, or positive payoff externalities. That result resembles ours,
but in a very different context. For clustering arising from coordination failures, see [27].
For effects of positive network externalities see [30], and for informational spillovers see [11],
and [9].
Or players may prefer to be neither first nor last; that general situation where rewards
depend on the players’ ordinal rank of timing action is modeled by [33] and by [2]. We too
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have order matter, but not by assumption but because customers have preferences over when
they are served, and the order in which a customer arrives affects when he is served.
3 Model
Consider n customers (jobs) requiring service from a single server. The service of customer
i takes one unit of time. The ideal service start time for customer i = 1,...,n, is d (due
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date). If customer i commences service at time s then he incurs a cost of
c (s) = β(s−d )+ +γ(d −s)+, i = 1,...,n , (1)
i i i
where β is the lateness cost and γ is the earliness cost.
Of course, a customer’s effective time of admittance into service is a function ofhis arrival
time and of all other arrival times. We therefore consider the non-cooperative game in which
customers simultaneously select their arrival times. The server admits customers according
to a FCFS regime. If multiple customers arrive at the same instant then they are admitted
into service in uniform random order. Let t = (t ,...,t ) be an arrival schedule (or profile),
1 n
and denote the expected service start time of customer i by s (t). The expected service start
i
times satisfy the following dynamics:
1
s (t) = t ∧ max s (t)+ 1 , i = 1,...,n , (2)
i i j:tj<ti j 2 {tj=ti}
j6=i
X
where 1 is the indicator function on condition A.
A
From now on we assume a homogeneous due date, d = 0, ∀i = 1,...,n. In Section 6.4
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we examine the consequences of heterogeneous due dates.
4 Social optimization
A central planner seeking to minimize the total cost,
n
c (s (t)) ,
i i
i=1
X
can achieve any possible sequence of service start times by setting arrival times at a distance
of at least 1 from each other. Therefore the social optimization problem is
n
min [β(s ∨0)−γ(s ∧0)] ,
i i
(s1,...,sn)
i=1
X
s.t. s ≥ s +1, i = 2,...,n .
i i−1
Note, however, that the identical sequence of start times can be achieved by several
arrival profiles, for example two customers arriving at the same instant will yield the same
sequence as that of one arriving exactly a unit after the other.
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We next show that with a common due-date the optimal service sequence is any sequence
with no holes such that 0 is the β percentile. This result arises because the objective
β+γ
function is the sum of absolute deviations from zero with different weights for negative and
positive values. Proposition 1 is a standard result in single machine sequencing (e.g. [7]).
Nevertheless, we provide the statement and proof for our specific formulation for the sake of
completeness and for easy comparison to the subsequent equilibrium analysis.
Proposition 1. Let s˜:= nβ . A sequence of service start times is optimal if and only if
β+γ
s = s +1, i = 2,...,n ,
i i−1
and
1. if s˜∈ then
N
s ∈ [−s˜,−s˜+1] ,
1
2. if s˜∈/ then
N
s = −⌊s˜⌋ .
1
Proof. Without loss of generality we assume s ≤ s ≤ ··· ≤ s . Since d = 0 for all
1 2 n i
i = 1,...,n, then (1) yields
c (s) = |s|(β1 +γ1 ) .
i {s>0} {s<0}
Clearly, an optimal service sequence has no holes, i.e. satisfies
s = s +1 = s +i−1, ∀i ≥ 2 .
i i−1 1
Hence there is a single decision variable: s . Denoting i := max{i : s ≤ 0}, the cost
1 0 i
function is
n i0
c(s ) = β(s +(i−1))− γ(s +(i−1))
1 1 1
i=Xi0+1 Xi=1
n i0
= β(n−i )−γi s + β(i−1)− γ(i−1) .
0 0 1
(cid:0) (cid:1) i=Xi0+1 Xi=1
Note that i depends on the choice of s . Therefore, the cost function is piecewise affine
0 1
with respect to s , with the slope determined by i : β(n−i )−γi . The slope is negative
1 0 0 0
if i > n β , positive if i < n β , and zero if i = n β (in case this is indeed an integer).
0 β+γ 0 β+γ 0 β+γ
Therefore, if nβ ∈ then any s satisfying
β+γ N 1
nβ
i = max{i : s +i−1 ≤ 0} = ,
0 1
β +γ
or equivalently,
nβ nβ
− ≤ s ≤ − +1 ,
1
β +γ β +γ
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is optimal. Otherwise, if nβ ∈/ then the objective function has a global optimum that
β+γ N
satisfies
nβ nβ
< i = max{i : s +i−1 ≤ 0} < +1 ,
0 1
β +γ β +γ
and s +i −1 = 0. Hence, we conclude that i = nβ , and lastly,
1 0 0 β+γ
l m
nβ nβ nβ
s = − +1 = − −1+1 = .
1
β +γ β +γ β +γ
(cid:24) (cid:25) (cid:22) (cid:23) (cid:22) (cid:23)
Corollary 2. If β = γ then a sequence of service start times s = s + i− 1 is optimal if
i 1
and only if zero is a median of the sequence.
5 Equilibrium analysis
An arrival profile t = (t ,...,t ) is a pure-strategy Nash equilibrium if
1 n
′ ′
c (s (t)) ≤ c (s (t ,...,t ,...,t )), ∀t ∈ , i = 1,...,n,
i i i i 1 i n i
R
where s () is given by (2).
i
5.1 Two-customer game
Consider first a two-customer example with symmetric deviation penalties: n = 2 and
β = γ. In this case t = t = −1 is the unique pure strategy equilibrium with both
1 2 2
customers incurring a cost of β. No customer has an incentive to deviate because arriving
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during − 1, 1 will cost exactly β, and arriving later or earlier than this interval is clearly
2 2 2
more costly. The uniqueness can be derived by considering the best-response function to any
(cid:0) (cid:3)
arrival t by the other customer (we use the notationt− to indicate that arriving momentarily
before the other customer is optimal):
0, t > 0,


 t−, −1 < t ≤ 0,
 2



b(t) =  −1, 1 , t = −1,
 2 2 2


(cid:2) (cid:3)
(t,1+t], −1 ≤ t < −1
 2




 0, t < −1.







The best-response is not necessarily unique and the pair of arrival times (t ,t ) is an equi-
1 2
librium if t ∈ b(t ) and t ∈ b(t ). Indeed, t = t = −1 is the only pair that satisfies this
1 2 2 1 1 2 2
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condition. This is illustrated for symmetric strategies in Figure 1. We conclude that the only
stable pair is −1 for both customers. Furthermore, the service start times in equilibrium are
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socially optimal.
b(t)
t
1
b(t) = 1+t
0.5
b(t) = 0
t
-1 -0.5 0 0.5 1
b(t) = 0
-0.5
b(t) = t−
te = −1
2 -1
Figure 1: Example: n = 2 and β = γ. Best response to the arrival time t of the second
customer. The only fixed point is the discontinuity point at t = −1. The solid red line and
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the dotted region is the best response function b(t).
5.2 General equilibrium properties
We consider next the equilibrium for any finite number of homogeneous customers and any
β,γ > 0. It turns out that the set of pure-strategy equilibria includes only symmetric equi-
libria, and is given by an interval of arrival times τ = (t,t), i.e. all customers simultaneously
arriving at time t is an equilibrium for every t ∈ τ. Before stating the main result we prove
several useful lemmas. All proofs are given in the Appendix.
Lemma 3. Any equilibrium arrival profile satisfies the following:
(a) The first customer arrives at some t < 0 and the last customer enters service at some
a
t > 0.
b
(b) The server operates continuously during the interval [t ,t ].
a b
Lemma 4. There is no asymmetric strategy equilibrium.
Lemma 5. There is no symmetric mixed-strategy equilibrium.
The conclusion of this section is that any Nash equilibrium is pure and symmetric, that
is, it is given by a single arrival time for all customers. In the following we will characterize
all such equilibria and explore when the socially optimal solution is also an equilibrium.
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5.3 Symmetric equilibrium
If all customers arrive at time t < 0 then the expected cost for each of them is determined
by the uniform random ordering, resulting in a cost of
n−1
t+i
β1 −γ1 .
{t+i≥0} {t+i<0}
n
i=0
X (cid:2) (cid:3)
If properties (a) and (b) of Lemma 3 are satisfied, then a single customer has two reasonable
options to deviate: arriving before everyone else and incurring a cost of at least −γt, or
obtaining service after all others have completed their service with a cost of at least β(t+
n−1). The latter can be achieved by arriving at any time during the interval (t,t+n−1].
Hence, the expected cost of choosing time s when all others play t is
−γs, s < t,

c(s;t) =  1 β n−1 (t+i)−γ it (t+i) , s = t, (3)
 n i=it+1 i=0


h i
P P
βmax{t+n−1,s}, s > t,






where i := max{i : t+i <0}.
t
For t to be a symmetric equilibrium the costs associated with deviating from t must
exceed the expected cost associated with arriving at t with all the others. An immediate
symmetric equilibrium te is given by
β
−γte = β(te +n−1) ⇔ te = −(n−1) .
β +γ
This is an equilibrium because from (3) we have that
1 1 n−2 ite 1
c(te;te) = − γte + β (te +i)−γ (te +i) + β(te +n−1)
n n n
" #
i=Xite+1 Xi=1
≤ β(te +n−1) = −γte ,
as every element in the interior sums is smaller than the first and last elements (the start
times are closer to zero), and therefore neither arriving before nor after all others benefits
any single customer.
The above equilibrium is unique for n = 2 but not in the general case. For example,
suppose that all arrive at te + ǫ for some small ǫ > 0. If n > 2 then by continuity of the
cost function both extremal costs exceed the expected cost obtained by arriving at te. This
behaviour is illustrated for an example with n = 5 in Figure 2.
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4
β(t+n−1)
3
c(t;t)
2
1
−γt
0 t
-4 -3 t te = −2 t -1 0
Figure 2: The cost of all customers arriving simultaneously at t, i.e. c(t;t), compared with
the cost a single customer can obtain by either deviation: −γt by arriving a moment before,
or β(t + n − 1) by arriving after all are served. Arriving at t is a best response to all
others arriving at t along the interval (−2.66,−1.33) where both deviations are more costly.
Example parameters: n = 5, β = γ = 1.
We are now ready to fully characterize all possible equilibria.
Proposition 6. Let t be the unique solution of β(t+n−1) = c(t;t) and t the unique solution
of −γt = c(t;t). The set of all equilibria is given by the pure and symmetric strategies of all
customers arriving together at time t, such that
t ∈ τe = [t,t] ⊂ (−(n−1),0) .
Proof. By Lemma 3a, t is not an equilibrium if t ≥ 0 or t ≤ −(n − 1). The equation
β(t+n−1) = c(t;t) has a solution t ∈ (−(n−1),0) because by (3), c(0;0) < (n−1)β and
c(−(n−1);−(n−1)) > 0. The cost function (3) is piecewise linear with respect to t, with
a slope of
β(n−1−i )−γi
t t
a(t) = ,
n
where i := max{i : t+i < 0} ≥ 0. Further, t ∈ (−(n−1),0) implies i ∈ (0,(n−1)) and
t t
therefore a(t) < β, which in turn implies that the solution t of β(t+n−1) = c(t;t) is unique.
Similarly, as c(0;0) > 0, c(−(n−1),−(n−1)) < (n−1)γ and |a(t)| < γ we conclude that
−γt = c(t;t) admits a unique solution, t. Furthermore, β(t+n−1) ≥ c(t;t) is satisfied for
any t ≥ t, and −γt ≥ c(t;t) is satisfied for any t ≤ t. We conclude that the interval τ is the
set of all pure strategy symmetric equilibria. The interval is not empty because we already
saw that te = −(n−1) β is always an equilibrium.
β+γ
Observe that for n = 2 there is a unique solution te = − β , and in particular te = −1
β+γ 2
for β = γ, as was shown in Section 5.1. Proposition 1 characterized the socially optimal
sequence of service start times, which can be achieved, for example, by a symmetric strategy
ofallarriving ats˜, i.e., attheoptimalfirst service time. Denotethissocially-optimalstrategy
by t∗ := s˜. It turns out that the social optimum is often an equilibrium, but not always, as
is illustrated by an example in Figure 3. We next give necessary and sufficient conditions
for the socially optimal solution to be an equilibrium, with the proof given in the appendix.
9
β(t+n−1)
10
8
6
c(t;t)
4
2 −γt
0 t
t∗ = −4 t te -3 t -2
Figure 3: The cost of all customers arriving simultaneously at t, i.e. c(t;t), compared with
the cost a single customer can obtain by either deviation: −γt by arriving a moment before,
or β(t+n−1) by arriving after all are served. Arriving at t is a best response to all others
arriving at t along the interval τe = (−3.58,−2.67) where both deviations are more costly.
However the minimal total cost is attained at t∗ = −4 as c(t;t) is an increasing function.
Example parameters: n = 5, β = 6, γ = 1.
Proposition 7. There exists a socially optimal symmetric arrival time that is also an equi-
librium if and only if β ∈ 1,1− 1 .
β+γ n n
Proposition 7 implies th(cid:2)at equilib(cid:3)rium clustering of arrival times is often efficient for
society. Specifically, when n is not very small there is a socially optimal equilibrium for
almost all values of β and γ. This means that as long as the penalty function is fairly
symmetric there is a socially optimal equilibrium. But when the penalty is heavily skewed
in one direction the equilibrium arrivals are too early (small β ) or too late (large β ).
β+γ β+γ
This is illustrated in Figure 4. Furthermore, it should be pointed out that even if for many
parameter values there is a socially optimal equilibrium, welfare under most of the equilibria
is less than under the socially optimal solution. In particular, the price of stability, which
is the ratio between the social optimum and the best equilibrium, is typically one, whereas
for n > 2 the price of anarchy, which is the ratio between the social optimum and the worst
equilibrium, exceeds 1.
β
β+γ
1
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0 n
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Figure 4: Range of parameter values (red dotted lines) such that there is a socially optimal
equilibrium.
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